My hope is to compile the "hidden" statistics of perks. Not to rank them according to best or worst, but to reveal the things about them that work better than the text implies, or that do less than the text implies. I will seek to update the thread as much as possible to incorporate the responses that people give. I will also continue to add more to the OP as I return to the game.
Perks with Hidden Advantages:
Scrapper Level 1
In spite of the text for both levels of scrapper, the first level of scrapper currently returns all the same rare components as the second level, just fewer and from less mod types.
Blitz
Blitz can simultaneously carry the sneak critical, weapon and Ninja bonuses for as many strikes as can be completed in a single VATS run (much like with other combat techniques and perks bonuses.) In the case of having both levels of Blitz, this means that you can melee a crowded room of separated enemies while still applying the full sneak bonuses to all of them. This isn't so much a "hidden" attribute of Blitz, specifically, but of several perks. Blitz can take the most advantage of the phenomenon because of the 10x damage multiplier from Ninja.
Sneak
The additional bonuses of levels 2-4 of sneak are applied to your active companion. This means that they won't set off mines or ground-based traps, either. The bonuses are not reliably applied to third party npcs or temporary companions, even if they are considered essential or critical to the mission.
Perks with hidden Disadvantages
Killshot
Ehrm... this probably belongs in the other category. But for now, until the next patch, Killshot returns a 2000% bonus to headshots, rather than 20%. This effectively breaks the combat of the game, which many would consider a disadvantage. Understandably, some would see it as an advantage. It is not listed in your Perk Tree, as it is a companion perk, acquired from MacReady (found in Goodneighour.) It cannot be acquired accidentally, as there are two companion missions that are required before the perk can be activated.
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I'll take some time to look over the perk tree later today and pull some more examples to expand the list.
I would appreciate any submissions anyone has, especially of perks with bugs that alter their expected performance.